When you are done registering time, you can save your project. Tomorrow you just open your project file again and continue where you left off today. If you want to continue the time registration while TimeTraces is closed, then you do not have to stop the time registration first before closing your project. If you do not stop the time registration on a project before closing it, all time will be registered correctly when you re-open the project later on.
The project(s) you had open last time you used TimeTraces will be reopened when you start up TimeTraces again. For your convenience, TimeTraces also keeps a list of the projects you recently worked on (under the "Project \ Open recent" menu); the maximum number of projects displayed in this menu can be set in the "Settings" window. Of course you can also explicitly open an existing project by browsing to it, using "Project \ Open..." (or by pressing the keys "Ctrl" and "O" from the main window).
TimeTraces can also start up when your computer starts. This way, your current project will always be loaded and ready to use when you start your computer. To enable this option, go to the "Settings" window (choose "Options \ Edit settings..." from the menu).